Abstract
We study the existence and attractivity of solutions for fractional evolution equations with Riemann-Liouville fractional derivative. We establish sufficient conditions for the global attractivity of mild solutions for the Cauchy problems in the case that semigroup is compact.
1. Introduction
Fractional differential equations have gained considerable importance due to their application in various sciences, such as physics, mechanics, chemistry, and engineering. In recent years, there has been a significant development on ordinary and partial differential equations involving fractional derivatives; see the monographs of Podlubny [1], Kilbas et al. [2], Diethelm [3], Tarasov [4], and Zhou [5, 6] and a series of papers [7–28] and the references cited therein.
Recently, Zhou [7], Chen et al. [19], Losada et al. [20], and Bana and O’Regan [21] investigated the attractivity of solutions for fractional ordinary differential equations and integral equations. On the other hand, the existence theory of solutions for time fractional evolution equations has been investigated intensively by many authors; for example, see Kim et al. [16], Bazhlekova [22], Wang et al. [23], Zacher [24], and Zhou et al. [25]. However, to the best of our knowledge, there are no results on the attractivity of solutions for fractional evolution equations in the literature.
Consider fractional evolution equation with Riemann-Liouville derivative:where is Riemann-Liouville fractional derivative of order , is Riemann-Liouville fractional integral of order , is the infinitesimal generator of a -semigroup of bounded linear operators in Banach space , is a given function satisfying some assumptions, and is an element of the Banach space .
In this paper, we initiate the question of the attractivity of solutions for Cauchy problem (1). We establish sufficient conditions for the global attractivity for mild solutions of (1) in the case that semigroup is compact. These results essentially reveal the characteristics of solutions for fractional evolution equations with Riemann-Liouville derivative. More precisely, integer order evolution equations do not have such attractivity.
2. Preliminaries
In this section, we firstly recall some concepts on fractional integrals and derivatives and then give some lemmas which are useful in next sections.
Let and . The Riemann-Liouville fractional integral is defined by where denotes the convolution, and in case , we set , the Dirac measure is concentrated at the origin. For , the Riemann-Liouville fractional derivative is defined by
The Wright function is defined by It is known that satisfies the following equality:
We give the following definition of the mild solution of (1).
Definition 1 (see [5]). By the mild solution of the Cauchy problem (1), we mean that the function satisfies where
Definition 2. The mild solution of the Cauchy problem (1) is attractive if tends to zero as .
Suppose that is the infinitesimal generator of a -semigroup of uniformly bounded linear operators on Banach space . This means that there exists such that where be the space of all bounded linear operators from to with the norm , where and .
Proposition 3 (see [5]). For any fixed , is linear and bounded operator, that is, for any ,
Proposition 4 (see [5]). is strongly continuous, which means that, for and , we have
Proposition 5 (see [5]). Assume that is compact operator. Then is also compact operator.
Let be a real Banach space, : with the norm . It is easy to see that is a Banach space.
We need also the following generalization of Ascoli-Arzela theorem, which one can find in [29].
Lemma 6. The set is relatively compact if and only if the following conditions hold: (i)For any , the function in is equicontinuous on .(ii)For any , is relatively compact in .(iii) uniformly for .
Theorem 7 (Schauder fixed point theorem). Let be a nonempty, closed, and convex subset of the Banach space and let be completely continuous; then has a fixed point in .
Lemma 8 (see [26]). If , then where
3. Some Lemmas
In this paper, we always suppose that the operator generates a compact -semigroup on ; that is, the operator is compact for .
Let with the norm Then is a Banach space by the similar proof of [27, Lemma ].
For any , consider the operator defined by where It is easy to see . It is clear that is a mild solution of (1) in if and only if there exists a fixed point , such that holds.
For any , let Then, . Define an operator as follows: where
In this section, we always suppose that the following condition holds:(H1)There exist and , such that , for any .
Let . Then there exists such that Let
Before obtaining our main results, we firstly prove some lemmas as follows.
Lemma 9. Assume that (H1) holds. Then, is equicontinuous and uniformly for .
Proof. It is clear that is a nonempty, closed, and convex subset of .
Claim I. is equicontinuous.
For , , we have For any and , we have Hence, is equicontinuous.
Claim II. is equicontinuous. For any , let , . Then , where which is nonempty, closed, and convex.
Since and is given, then there exists enough large, such that For , in virtue of (H1) and (26), we get For , we have For , we have where It can deduce that as directly. Indeed, Note that and the map is integrable for and , then by Lebesgue dominated convergence theorem, we have which implies that as .
For given small enough, from the condition (H1), we get where By Proposition 4, we have that as . Similar to the proof that , tend to zero, we get and as . Thus, we get that tend to zero independently of as , .
Similar to (24), it is easy to proof that as .
For , note that if , then and , from the above argument, we obtain Therefore, it is clear that independently of . Thus, is equicontinuous..
Claim III. uniformly for . For any , by (H1) and Proposition 3, we have We multiply the above inequalities at both sides with the factor , then and, therefore, which shows that uniformly for . The proof is completed.
Lemma 10. Assume that (H1) holds. Then maps into itself and is continuous.
Proof.
Claim I. maps into . For any , let , . Then .
For , together inequity (21) and yield that Then, for , from (39), we have which deduce that for .
Claim II. is continuous. For any , with , let , Then .
In addition, we have By (H1) and the continuity of on any compact subsets uniformly, then we get Therefore, for a.e. , via (H1). In addition, the function is integrable for and . Hence, by Lebesgue dominated convergence theorem, we obtain Then, for , which yields that as .
On the other hand, let be given, fixed enough large with replaced such that (26) holds. Then, for , by virtue of (26) and (27), we have Therefore, it is obvious that as Combined with the above statement, It can imply that uniformly on as ; that is, is continuous.
4. Main Results
Theorem 11. Assume that is compact, and the condition (H1) holds. Then the Cauchy problem (1) admits at least one attractive solution.
Proof. Obviously, is a mild solution of (1) in if and only if is a fixed point of in , where . Thus, it is sufficient to show that has a fixed point in . By Lemma 9, we know that is equicontinuous and uniformly for . It remains to verify that, for any , is relatively compact in according to Lemma 6. Obviously, is relatively compact in . Let be fixed. For every and , define an operator on as follows: Since is compact for , by Proposition 5, we know that is compact. In addition, from the compactness of , we obtain that the set is relatively compact in for any and for any . For every , we have Therefore, the set is closed to an arbitrary compact set. As a result, the set is also relatively compact set in for . By Lemma 6, we know that is a relatively compact set. On the other hand, by Lemma 10, we know that maps into itself and is continuous. Hence, is a completely continuous operator. Therefore, according to Schauder fixed point theorem, there exists at least one fixed point such that holds. Let ; then and, therefore, is a mild solution of (1).
Noting that, for any , using the condition (H1), we have which yields that as . Thus, the solution is attractive.
Theorem 12. Assume that is compact, and function satisfies the following condition: (H2)There exist , and , such that , for any . Then the Cauchy problem (1) admits at least one attractive solution.
Proof. Let . Then there exist constants and such that Let For any , let , . Then , where By (H2) and Lemma 8, similar to (39), we get for , together inequity (52) and yield that Then, for , from (55), we have which deduce that for .
The remaining part of the proof is similar to that of Theorem 11, and we omit it.
5. Example
Example 1. Let Consider the following Cauchy problem for fractional partial differential equations: where , is a given function, .
We define an operator by with the domain Then generates a compact, analytic, self-adjoint -semigroup .
Let ; that is, , , . And the function is given by Then the Cauchy problem (58) can be rewritten as the following format in :
We can take and . Then, (H1) is satisfied. According to Theorem 11, problem (58) has at least one attractive solution.
However, for the following integer order differential evolution equations, As the argument described above, (62) can be rewritten as
Then (63) has an equivalent representation:with as . This result shows that fractional evolution equations with Riemann-Liouville derivative have the global attractivity, whereas the integer order evolution equations do not have such attractivity.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work was supported by the National Natural Science Foundation of China (no. 11671339).